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User blog:Vendellium/Let's Count Sheep! ~ Bar Dream Graphs
The Sleepy Calculus The Sleepy Calculus (SC) is a name to refer to the symbols of dream notation used to represent dreams and sheep in a mathematical way and the rules of syntax joining these symbols, which will be developed in following blog posts. The name was chosen because Dream Mathematics has already been taken. Bar Dream Graphs Bar dream graphs are a way to keep track of sheep within the nested dreams of SC. These graphs take the form of rectangles which are placed above or below each other. An arrow to the left or right of the graph indicates in which direction the dreams are to be nested. A solitary individual is denoted by one rectangle: If this individual has a dream of one sheep then their dream is represented as a rectangle above or below the individual's rectangle. Here the dream is represented as a bar above the individuals rectangle so the up arrow indicates the dreams are higher nested the farther North you go. Let us the dream this person has is of a sheep. Then logically, that sheep may have a dream of it's own, so that the dreamed-sheep's dream is the third bar up in our nesting of dreams. Then, that sheep may have a dream of it's own where a sheep is dreamed of, and so on, till you get graph like this one: Here the topmost bar represent the dream of the original individual's dream-sheep's dream-sheep's dream-sheep's dream-sheep's dream-sheep's dream-sheep's dream-sheep's dream-sheep's dream. Of course, we can continue in this way until we have as many layers as you want, but it wouldn't be very interesting, only tedious. So instead, we'll make some simple rules for our dreaming individual and sheep, and see where it takes us. First, let us say that every unit of time T the person will dream another sheep. At time 1 the person dreams of one sheep - at time 2 the person dreams of 2 sheep - at time 3 the person dreams of 3 sheep, and so on. Second, every sheep that is dreamed of will dream of it's own sheep at a rate of one sheep per unit of time. This is a simple set of rules, but how does it look according to our dream graphs? Well first you have the person, with no dreams, at time 0: Then, they dream of a single sheep at time 1: It's the same as before. But at time 2 things change. Recall that according to our rules the sheep the person dreamed of will now dream of it's own sheep, and the person will then dream of a second sheep in addition to the first one. So two new sheep are added in total: At time 3 our dream graph looks like the following: At time 4 our dream graph looks like this: So it continues, with each successive graph growing more complex than the last. Except...hold on...something's odd about these graphs... let's count how many bars are in each layer of these graphs... There is 1 bar. The number of bars are 1 and 1, for 2 layers. The number of bars are 1, 2, and 1, for 3 layers. The number of bars are 1, 3, 3, and 1, for 4 layers. The number of bars are 1, 4, 6, 4, and 1, for 5 layers. Now where might these numbers appear? Let's see... This is Pascal's triangle. Each number is calculated by the summation of the two numbers above it to the left and right. If there is a blank space it is assumed to be a 0. The number atop this triangle of summation is 1. Evidently then, there is a definite correlation between counting sheep in the way I did and Pascal's triangle. We should expect that at time 5 our dream graph from layer to layer would have 1, 5, 10, 10, 5, and 1 bars - and if you work it out - it does. So, to calculate how many bars in a particular row m'' there are at time ''n in these dream graphs just find the number in Pascal's triangle that corresponds to mth number in the nth row. But if you'd rather not write out the entire triangle up until that point to find it, is there an equation you can use that would help you? Yes there is! The derivation of such an equation will be given soon. Category:Blog posts